Evolutionary game analysis of low-carbon technology innovation diffusion under PPP mode in China

With the Public–Private Partnership (PPP) mode playing an increasingly important role in the investment of infrastructure, promoting the diffusion of low-carbon technology innovation in PPP projects not only helps to reduce the carbon emissions of infrastructure, but also plays an important demonstration role in the low-carbon transformation of construction industry. In order to clarify the evolutionary mechanism of low-carbon technology innovation diffusion among stakeholders under the unique payment mechanism of PPP projects in China, the paper builds two different evolutionary game models respectively under government payment mechanism and consumer payment mechanism based on evolutionary game theory. The evolutionary relationship between the choice of behavior strategies and the change of influencing factors under different payment mechanisms are analyzed by numerical simulation. It is found that under government payment mechanism, the regulation cost of government to promote low-carbon technology innovation and the punishment of the superior regulatory authority for non-regulation behavior are the important factors affecting government’s behavioral strategies. The low-carbon technology innovation cost of social capital and the intensity of government subsidy and punishment are the main factors affecting social capital’s behavioral strategies; Under consumer payment mechanism, consumer becomes the ultimate payer of low-carbon products, and the income they get from purchasing low-carbon products and the subsidy provided by the government become the main factors determining consumer’s behavioral strategies. Finally, from the perspective of government, social capital and consumer, countermeasures and management implications are put forward to effectively promote the diffusion of low-carbon technology innovation under different payment mechanism in PPP projects.


Introduction
In the new era, low-carbon technology innovation and green concept as the guidance are important connotations of high-quality economic development [1,2] and how to reduce carbon emissions has become a hot issue in various industries [3][4][5]. According to statistics published by the UN Global Compact, while providing basic services to the public, infrastructure such as construction, transportation and other industries also contribute about 70% of global carbon emissions. Meanwhile, the Global Infrastructure Hub predicts that global demand for public infrastructure investment will reach $94 trillion by 2040. It is expected that such a huge amount of infrastructure will bring more carbon emissions and it is urgent to carry out lowcarbon transformation and sustainable development reform in the field of infrastructure. As a new investment approach of infrastructure, PPP can not only save the costs for the government, but also inject the private sector efficiency in the government sector domain [6]. In 2016, the Chinese government proposed to actively use PPP mode to support the transformation and development of low-carbon economy, and now it has been widely used in 19 industries such as municipal engineering, transportation and environmental governance, etc. [7,8]. By the end of 2021, the PPP project database of the Ministry of Finance of China has 10,243 projects with a total investment of 16.2 trillion CNY. As shown in Table 1, among the top five industries, the two major infrastructure industries such as municipal engineering and transportation accounted for more than 50%. Furthermore, it is obvious that low-carbon technology innovation requires the participation of all stakeholders in the whole life cycle, rather than a single enterprise or a single stage [9]. The implementation process of infrastructure under PPP mode covers all types of participants in the whole life cycle [10] and it has natural advantages in leading the collaborative promotion of low-carbon technology innovation through the whole process of design, construction and operation with social capital as the core. Due to the advantages of PPP mode and its increasing important role in infrastructure, it can be determined that promoting low-carbon technology innovation in PPP projects is not only conducive to reducing the overall energy consumption of PPP products, but also plays an important demonstration role in promoting the sustainable development of infrastructure.
Low-carbon technology is a relatively broad concept. For construction industry, low-carbon technology refers to the general term of a class of technologies that can reduce the use of fossil energy, improve energy efficiency and reduce carbon dioxide emissions in the process of construction and operation [11]. As Blaut put forward that the role of innovation diffusion is more important than innovation itself [12], low-carbon technology innovation can realize its economic value only when it is adopted and commercialized by enterprises. Due to the diffusion process of low-carbon technological innovation is a dynamic process in which multiple participants interact with each other, and the strategic choices made by different participants based on their own interests maximization will conflict, so the decision-making process of multiple participants can be better understood from the perspective of evolutionary game theory which has been widely used for behavioral strategy evolutionary mechanism analysis [13]. Some scholars have researched the diffusion of low-carbon technology innovation in construction industry based on evolutionary game theory. The majority of previous research has focused on the main body and the affecting factors of low-carbon technology innovation diffusion. It believed that government, enterprise and consumer are the main body that affects the diffusion of low-carbon technology [14][15][16]; Government subsidy, innovation cost and consumer willingness are the main factors affecting the stakeholders' behavior [17][18][19][20]. The existing literature provides a good reference for the study. But the diffusion of low-carbon technology innovation in PPP projects is much more complicated. Existing researches on the diffusion of low-carbon technology innovation are mainly based on consumer payment, which namely "who benefits who pays" [21,22]. The products of PPP projects mainly in the area of infrastructure and public service that completely rely on consumer payment mechanism is not reality, so the payment mechanism of PPP projects in China mainly includes three categories: government payment, consumer payment as well as feasibility gap subsidy. Existing studies are based on consumer payment, without considering other payment mechanisms of PPP projects. The stakeholders of PPP projects may choose different behavioral strategies for low-carbon technology innovation under different payment mechanisms [23]. The key issue to promote the diffusion of low-carbon technology innovation in PPP projects is how to clarify the behavior evolutionary mechanism of low-carbon technology innovation among stakeholders under different payment mechanisms and explicit the main factors affecting the stakeholders' behavior. Therefore, in order to clarify the evolutionary mechanism and influencing factors of low-carbon technology innovation diffusion under different payment mechanisms in PPP projects, this paper builds two evolutionary game models of low-carbon technology innovation composed of different stakeholders under different payment mechanisms of PPP projects, analyzes the evolutionary process of system equilibrium strategy under different circumstances, and further studies the influence of relevant factors on system equilibrium by means of numerical simulation experiments. The following research questions will be answered in this study.
• What is the evolutionary mechanism of low-carbon technology innovation under different payment mechanisms in PPP projects?
• What are the main influencing factors of behavior evolutionary for low-carbon technology innovation in PPP projects?
• How to promote the diffusion of low-carbon technology innovation in PPP projects?
The paper studies the evolutionary mechanism of low-carbon technology innovation diffusion under the unique payment mechanism of PPP projects, the research results will enrich the research perspective of low-carbon technology innovation diffusion and the application of evolutionary game theory on PPP projects. In addition, the paper analyzes the behavior evolutionary trend of different stakeholders under different payment mechanisms and puts forward management implications, it will help to improve the diffusion of low-carbon technology innovation in PPP projects and promote the sustainable development of infrastructure industry. This study is structured as follows: Firstly, we review the relevant literature and point out our research opportunities. Secondly, we build the evolutionary game model for low-carbon technology innovation of PPP projects and analyze the system equilibrium strategy. Finally, we analyze the key influencing factors of behavior strategy by numerical simulation and put forward some management suggestions on the diffusion of low-carbon technology innovation in PPP projects.

Low-carbon technology innovation diffusion
Low-carbon technologies are mainly zero-emission or low-emission renewable energy technologies (including wind and solar energy), as well as carbon emission reduction technologies and carbon capture and storage technologies [11]. Low-carbon technology innovation refers to the process of unlocking the original technological economic system through the transformation of technological paradigm [24]. As the follow-up process of technology innovation, the concept of technology innovation diffusion can be traced back to the innovation theory founded by Schumpeter. He believed that successful technology innovation would not only bring monopoly profits to enterprises, but also cause competition among other enterprises to imitate, and such large-scale imitation of technology innovation is called technology innovation diffusion [25].
The traditional technology innovation diffusion theory provides an important theoretical basis for the research of low-carbon technology innovation diffusion [26]. Scholars at home and abroad have studied the behavior evolutionary of low-carbon technology innovation diffusion from different subjects of low-carbon technology innovation. 1) Between governments. Barrett and Kennedy analyzed the non-cooperative game of environmental decision-making between local governments in imperfect competition market [27,28]. Jingyu Wang and Anna Shi used the bargaining model in game theory to discuss the game behavior between local government and central government in the diffusion of low-carbon technology innovation [29]. 2) Between government, enterprise and consumer. Sheu et al. used three-stage game model to argue that the government should adopt green tax and subsidy policies to promote the development of low-carbon supply chain [30]. Guozhi Wang et al. analyzed the influence mechanism of low-carbon technology innovation input subsidy and carbon tax on enterprises' lowcarbon technology innovation behavior and their choice of behavior strategies [31]. Mingyue Wang et al. build evolutionary game model of green technology innovation among government, enterprise and consumer, analyzed the evolutionary process of the influence of participants' strategy choice on system equilibrium strategy under different circumstances [1]. 3) Between enterprises. Przychodzen et al. believe that the return on assets and equity can promote low-carbon technological innovation of enterprises [32]. Yingying Xu et al. and Hongjuan Zhang et al. explored the macroscopic diffusion phenomenon of enterprise clusters emerging from the micro-decision interaction mechanism of potential technology innovation adopters in the context of low-carbon economy by using complex network evolutionary game [33,34]. Yingying Shi et al. explored the dynamics of low-carbon technology diffusion among enterprises with an evolutionary game model on a two-level heterogeneous social network [35].
In infrastructure industries, in terms of the subjects of low-carbon technology innovation, some scholars have carried out studies on how to promote low-carbon and green production between government and enterprise [36][37][38]. And some scholars believe that consumers are not only the supervisor of the construction process, but also the final consumer of low-carbon products, which plays an important role in promoting the diffusion of low-carbon technology innovation in construction industry. Therefore, governments, construction enterprises and consumers are widely considered to be the main participants in low-carbon technology innovation of construction industry [14][15][16]. Secondly, the influence factors of low-carbon technology innovation for different participants are studied. The reputation cost, reward and punishment policy, low-carbon consumption compensation policy are considered to be the major influence factors of government; the cost of low-carbon, intensity of government regulation are considered to be the main influence factors of construction enterprise; the low-carbon consumption preference, consumption cost and consumption subsidy are considered to be the main influence factors of consumer [17][18][19][20].

Evolutionary game theory
Evolutionary game theory is an application of the mathematical framework of game theory to the dynamics of animal conflicts [39,40]. Different from game theory, evolutionary game theory which is based on the assumption of bounded rationality not only considers the limited rationality of individuals, but also dynamically adjusts their strategies through learning and imitation. It both well describes the dynamic evolutionary process among the core stakeholders and better explains the formation process of equilibrium [41,42]. The two most important concepts in evolutionary game theory are Evolutionary Stability Strategy (ESS) and replicator dynamics [43].
Evolutionary game theory provides a wide range of research tools, its application range from evolutionary biology extends to various fields such as natural science and social science, and is especially widely used in economics. The research in this area applies evolutionary game theory in following main contexts: 1) The evolution of social customs, traditions and norms. Peyton Young applied evolutionary game theory to analyze the relationship between the formation of social norms and social welfare [44]. Fudenberg applied evolutionary game theory to analyze social learning process [45].
2) The changes of economic system and the evolution of market environment. Bester and Guth applied evolutionary game theory to study the existence and evolutionary stability of human altruism in economic activities [46]. The evolution of market survival and market credit are both studied based on the theory [47,48].
3) The research of enterprise organization, technology innovation and strategy. Freidman and Fung applied evolutionary game to analyze the evolution of the organization mode of firms with and without trade based on American and Japanese firms [49]. Yongping Wang and Weidong Meng applied evolutionary game theory to study the evolution of cooperative competition mechanism of supply chain enterprises [50].
Due to the good expression of the behavior evolutionary mechanism, evolutionary game theory has been widely applied to study the dynamic mechanism of low-carbon and green technology innovation diffusion. Scholars used evolutionary game theory to construct behavioral evolutionary models of low-carbon technology innovation among different subjects, and analyze the stability of system as well as the factors influencing strategies of different subjects. For instance, Jianpeng Zhou, Jingfeng Hu and Jingan Wang et al. used evolutionary game theory to discuss whether enterprises will implement low-carbon production under the guidance of government policies. They believed that it is uncertain for enterprises to choose low-carbon production strategies only by themselves. In order to promote the transformation of economic development mode to low-carbon, the government should have reasonable and effective policy arrangements for enterprises [51,52]. Suyong Zhang et al. developed the models of evolutionary game between governments and manufacturers, and analyzes the impacts of government policy on the decisions of manufacturers' emissions abatement behavior and the dynamic tendency of cap-and-trade market [53]. Nan Jiang et al. tried to develop a fractional-order game model of green and low-carbon innovation evolution in manufacturing enterprise [54] and Some other scholars have studied the low-carbon technological innovation of enterprise clusters by using evolutionary game theory [55]. In infrastructure industries, scholars mainly applied evolutionary game theory to study the evolutionary mechanism and influencing factors of low-carbon technological innovation in construction enterprises [22]. Some scholars also applied evolutionary game theory to study the technological innovation of prefabricated buildings [17].

Methodology
Evolutionary game theory which has been widely used for behavioral strategy evolutionary mechanism analysis is applied to research the evolutionary mechanism of low-carbon technology innovation diffusion in PPP projects. The research framework is shown in Fig 1. Firstly, the evolutionary game models of low-carbon technology innovation diffusion among stakeholders under different payment mechanisms in PPP projects are established.
Take government payment mechanism as example, the profit and loss of stakeholders (government and social capital) when they choose different strategies are determined, and then the evolutionary game model is formed by establishing the replication dynamic equations of each stakeholder. So does consumer payment mechanism.
Secondly, the models under different payment mechanisms are solved and the ESS of different stakeholders' behavior are analyzed respectively. Under government payment mechanism, the equilibrium points are obtained by solving the replication dynamic equation, and then the stability of different stakeholders' (government and social capital) behavior evolutionary is analyzed to determine the ESS point and stability conditions. Under consumer payment mechanism, evolutionary becomes a three-party game (government, social capital and consumer), the stable state of each stakeholder is obtained by solving the replication dynamic equation and the Lyapunov indirect method is used to judge the stability of the three-party game system.
Finally, Simulation software MATLAB R2018a is used to simulate the evolutionary process and analyze the main factors affecting low-carbon technology innovation diffusion in PPP project. Initial value is assigned to all parameters, and then the influence of each stakeholder's behavior evolutionary is observed by changing the value of parameters, and then the evolutionary mechanism of each stakeholder is summarized. At last, the suggestions are proposed according to the simulation results.

Evolutionary game model under government payment mechanism
Basic assumption and evolutionary game strategy analysis. Assumption 1: In PPP projects, government hands over the whole project process of financing, design, construction and operation to social capital who becomes the actual investor of PPP projects, while government assumes the role of supervision and service [56]. Although government no longer participates in the implementation of PPP projects, it has the responsibility to promote the diffusion of low-carbon technology innovation in PPP projects by establishing regulations. So the behavior strategies of government can be set as: {regulation, non-regulation}; Further, as the actual investor of PPP projects, social capital can effectively coordinate the diffusion of low-carbon technology innovation among all participants [57], so the behavioral strategies of social capital can be set as: {innovation, non-innovation}.
Assumption 2: In PPP projects, government and social capital show the characteristics of bounded rationality under the influenced and interfered by uncertain factors such as environment, policy and information [58]. For government, the probability of 'regulation' is set as x (0�x�1), and the probability of 'non-regulation' is equal to 1−x. For social capital, the probability of 'innovation' is set as y (0�y�1), and the probability of 'non-innovation' is equal to 1−y.
Assumption 3: Governments and social capitals will generate different benefits or expenses when they adopting different behavior strategies [59]. For instance, when government takes regulation strategy, it will pay regulation cost 'C1'; it will provide reward 'S1' to social capital that actively innovate in low-carbon technology; meanwhile, it will obtain some benefits 'R1' when social capital choose the innovation strategy, etc. The specific profit and loss indexes of each stakeholder are shown in Table 2. One stakeholder's behavior strategy will not change as other stakeholders' approach.
Based on the above assumptions, the profit and loss matrix of value among governments and social capitals is constructed as shown in Table 3. Table 3, we can get the expected value Eg1 of government taking strategy of 'regulation' and the expected value Eg2 of governments taking strategy of 'non-regulation', and they are shown in Eqs (1)-(2).

Construction and solution of evolutionary game model. According to
Then the average expected value of government is shown in Eq (3). Therefore, the replication dynamic equation of governments is shown in Eq (4).
Similarly, the replication dynamic equation of social capital is shown in Eq (5): Order dx dt ¼ 0 and dy dt ¼ 0, we can get five equilibrium nodes as (0,0), (1,0), (0,1), (1,1), (x � , y � ), in which x � ¼ C2À R1 S1þL1 ; y � ¼ L1þL2þL3À C1 S1þL1þL2 . The stability analysis of different subjects' behavior evolutionary. According to the method proposed by Friedman [60], the evolutionary stability of a two-dimensional dynamic system can be derived through local stability analysis of the Jacobin matrix of the system. The Jacobin matrix of the system is represented by J1 and shown as Eq (6).  C1 The cost paid by governments when they choose the strategy of 'regulation'

S1
The reward provided by governments when social capitals choose the strategy of 'innovation'

L1
The punishment imposed by governments when social capitals choose the strategy of 'noninnovation'

L2
The social loss that governments choose the strategy of 'non-regulation' and social capitals choose the strategy of 'non-innovation'

L3
The punishment imposed by superior supervisory authority when governments choose the strategy of 'non-regulation'

Social capital
R3 The income obtained by social capitals when they choose the strategy of 'innovation'

R4
The income obtained by social capitals when they choose the strategy of 'non-innovation'

C2
The cost paid by social capitals when they choose the strategy of 'innovation'

S1
The reward provided by governments when social capitals choose the strategy of 'innovation'

L1
The punishment imposed by governments when social capitals choose the strategy of 'noninnovation' https://doi.org/10.1371/journal.pone.0279493.t002 Among them,  Table 4. It can be seen from Table 4 that the value of tr J (a 11 +a 22 = 0) at the equilibrium point (x � , y � ) is not satisfied the condition tr J (a 11 +a 22 <0). Therefore, the point (x � , y � ) is not the ESS of the system and only points (0,0), (1,0), (0,1) and (1,1) will be considered. In addition, the values of both a 12 and a 21 at the four points are equal to zero, so the conditions for determining the ESS of the four points can be simplified as det J = a 11 a 22 >0 and tr J = a 11 +a 22 <0. That is to say, the values of a 11 and a 22 at the four points are negative at the same time.
It it obvious that low-carbon technology innovation needs the joint efforts of government and social capital in PPP projects. So we hope the system evolutionary stable culminate in point (1,1), that is, government chooses to create a good environment for promoting low-carbon technology innovation by regulating; while social capital chooses to carry out low-carbon technology innovation actively which would promote the sustainable development of construction industry. To achieve the above status, the system must meet the conditions S1+C1 −L3<0 and −S1−L1−R1+C2+R2<0 as shown in Table 5.

Evolutionary game model under consumer payment mechanism
Basic assumption and evolutionary game strategy analysis. Assumption 1. Under consumer payment mechanism, consumers become the payers of low-carbon products. So the behavioral strategies of consumers can be set as: {consumption, non-consumption}. Governments still bear the task of supervision and management [50], so the behavior strategies of governments also can be set as: {regulation, non-regulation}. Social capitals are still the main body of low-carbon technology innovation and the behavioral strategies of social capitals also can be set as: {innovation, non-innovation}.  Values of a 11 , a 12 , a 21 , a 22 https://doi.org/10.1371/journal.pone.0279493.t004 Table 5. Analysis on local stability of system equilibrium points. Assumption 2. In PPP projects, government, social capital and consumer show the characteristics of bounded rationality under the influenced and interfered by uncertain factors such as environment, policy and information [51]. So the probability of 'regulation' can be set as x (0�x�1), and the probability of 'non-regulation' is equal to 1−x. The probability of 'innovation' can be set as y (0�y�1), and the probability of 'non-innovation' is equal to 1−y. The probability of 'consumption' can be set as z (0�z�1) and the probability of 'non-consumption' is equal to 1−z.

Equilibrium point
Assumption 3: Governments, social capitals and consumers will generate different benefits or expenses when they adopting different behavior strategies. For instance, when government takes regulation strategy, it will pay regulation cost 'C1'; it will provide reward 'S1' to social capital that actively innovate in low-carbon technology and offer subsidy 'S2' to consumers who buy low-carbon products; meanwhile, it will obtain some benefits 'R1' when social capital choose the innovation strategy, etc. The specific profit and loss indexes of each stakeholder are shown in Table 6. One stakeholder's behavior strategy will not change as other stakeholders' approach. C1 The additional cost paid by governments when they choose the strategy of 'regulation'

S1
The reward provided by governments when social capitals choose the strategy of 'innovation'

S2
The subsidy provided by governments when consumers choose the strategy of 'consumption'

L1
The punishment imposed by governments when social capitals choose the strategy of 'noninnovation'

L2
The social loss that governments choose the strategy of 'non-regulation' and social capitals choose the strategy of 'non-innovation'

L3
The punishment imposed by superior supervisory authority when governments choose the strategy of 'non-regulation'

R3
The income obtained by social capitals when they choose the strategy of 'innovation'

R4 The income obtained by social capitals when they choose the strategy of 'non-innovation'
C2 The cost paid by social capitals when they choose the strategy of 'innovation'

S1
The reward provided by governments when social capitals choose the strategy of 'innovation'

L1
The punishment imposed by governments when social capitals choose the strategy of 'noninnovation'

L4
The compensation provided by social capitals when consumers choose the strategy of 'consumption' and social capitals choose the strategy of 'non-innovation'

R6 The income obtained by consumers when they buy traditional products
C3 The additional cost paid by consumers when they buy low-carbon products

S2
The subsidy provided by governments when consumers choose the strategy of 'consumption'

L4
The compensation provided by social capitals when consumers choose the strategy of 'consumption' and social capitals choose the strategy of 'non-innovation' Based on the above assumptions, the profit and loss matrix of value among stakeholders is constructed as shown in Table 7. https://doi.org/10.1371/journal.pone.0279493.t006

PLOS ONE
Evolutionary game analysis of low-carbon technology innovation diffusion under PPP mode in China Table 7, we can get the expected value E 0 g1 of governments taking strategy of 'regulation' and the expected value E 0 g2 of governments taking strategy of 'non-regulation', and they are shown in Eqs (7)- (8).

Construction and solution of evolutionary game model. Replicator dynamics equation and equilibrium point analysis of government. According to
Then the average expected value of government is shown as Eq (9): Therefore, the replication dynamic equation of government is shown as: The first derivative of x is equal to Eq (11).
When F(x) = 0 and d(F(x))/dxh0, governments' strategy of 'regulation' will be in a stable state.
Therefore, there will be F(x)�0 when z ¼ , there will be d(F(x))/dx| x = 0 <0, and x = 0 is the ESS of the system.
Replicator dynamics equation and equilibrium point analysis of social capital. According to Table 7, we can get the expected value E 0 p1 of social capitals taking strategy of 'innovation' and Table 7. The matrix of expected value of profit and loss among stakeholders.
Then the average expected value of social capitals is shown as Eq (14).
Therefore, the replication dynamic equation of social capitals is shown as Eq (15).
The first derivative of y is equal to Eq (16).
When F(y) = 0 and dðFðyÞÞ dy < 0, social capitals' strategy of 'innovation' will be in a stable state.

L4
, and all the values of y are in a stable state. When z < R4þC2À R3À xS1À xL1 L4 , there will be d(F(y))/dy| y = 1 <0, and y = 1 is the ESS of the system. When z > R4þC2À R3À xS1À xL1 L4 , there will be d(F(y))/dy| y = 0 <0, and y = 0 is the ESS of the system.
Replicator dynamics equation and equilibrium point analysis of consumer. According to Table 7, we can get the expected value E 0 c1 of consumers taking strategy of 'consumption' and the expected value E 0 c2 of consumers taking strategy of 'non-consumption', and they are shown in Eqs (17)- (18).
Then the average expected value of consumers is shown as Eq (19).
Therefore, the replication dynamic equation of consumers is shown as Eq (20).
The first derivative of z is equal to Eq (21).
When F(z) = 0 and dðFðzÞÞ dz < 0, consumers' strategy of 'consumption' will be in a stable state. Therefore, there will be F(z)�0 when y ¼ R5À R6À C3þL4þxS2

L4
, and all the values of z are in a stable state. When yi R5À R6À C3þL4þxS2 L4 , there will be d(F(z))/dz| z = 1 <0, and z = 1 is the ESS of the system. When yh R5À R6À C3þL4þxS2 L4 , there will be d(F(z))/dz| z = 0 <0, and z = 0 is the ESS of the system.

Evolutionary stability analysis
For the stability analysis of three-party evolutionary game, it is no longer applicable to judge the stability by the positive and negative of the determinant (trJ) and trace (detJ) of equilibrium Jacobian matrix [61]. In this study, Lyapunov indirect method is used to judge the stability of the three-party game system. Firstly, the Jacobian matrix and eigenvalues of the system are solved, and the Jacobian matrix of the system is expressed by J2 and shown as Eq (22).

Numerical simulation
In order to better analyze the evolutionary mechanism of low-carbon technology innovation under different payment mechanism, the study uses MATLAB R2018a to simulate the trend and influencing factors of the evolutionary game.

Numerical simulation under government payment mechanism
According to the above analysis, if the system converges to the ESS point (1,1), the following conditions need to be satisfied: S1+C1−L3<0 and −S1−L1−R3+C2+R4<0. The evolutionary process of low-carbon technology innovation among government and social capital is affected by multiple parameters including C1, C2, L1, L2, L3 and S1. In order to better reflect the actual situation, this paper uses data such as the statistical yearbook of the construction industry to calculate the parameters in the model. For example, in 2021, the average income of construction enterprises in China is 213 CNY/m 2 , we integer it to R4 = 2 hundred CNY/m 2 for the convenience of simulation. The government will punish construction companies with high carbon production. According to the current penalty for carbon emissions imposed by the environmental protection agency, we set L1 = 1 hundred CNY/m 2 . The average technological innovation investment of construction enterprises in energy conservation and emission reduction is 49.8 CNY/m 2 , and we integer it to C2 = 0.5 hundred CNY/m 2 . The government's fiscal subsidy for enterprises to carry out low-carbon production is 10 CNY/m 2 and we integer it to S1 = 0.1 hundred CNY/m 2 . Other parameters are determined based on comprehensive consideration of production data of construction enterprises, environmental compensation policies and social welfare. The initial values of each parameter are set to: x = 0.5, y = 0.5, R3 = 2.5, R4 = 2, C1 = 0.5, C2 = 0.5, L1 = 1, L2 = 1, L3 = 1, S1 = 0.1. By changing parameter values, the evolutionary mechanism of two behavioral strategies under different parameter changes is analyzed. The influence of C1 and L3 changes on the evolutionary of government's behavior. The evolutionary trend between government and social capital under the initial values of each parameter is shown as blue curves in Figs 2 and 3. Keep the other parameters unchanged, we can observe the evolutionary trend of government's behavior by changing the values of C1 and L3. In Fig 2, we can see that the convergence rate of x to the ESS point (1,1) is accelerated as the value of C1 goes down from 0.5 to 0.1. The simulation results show that reducing the cost of government's strategy of 'regulation' is helpful to enhance the willingness of government to promote low-carbon technology innovation in PPP projects. In Fig 3, we can see that the convergence rate of x to the ESS point (1,1) is accelerated as the value of L3 goes bigger from 1 to 1.4. The simulation results show that increasing the severity of punishment from superior supervisory authority will help increase the willingness of government to promote low-carbon technology innovation in PPP projects.
The influence of C2, L1 and S1 changes on the evolutionary of social capital's behavior. The evolutionary trend between government and social capital under the initial values of each parameter is shown as blue curves in Figs 4-6. Keep the other parameters unchanged, we can observe the evolutionary trend of social capital's behavior by changing the values of C2, L1 and S1. In Fig 4, we can see that the convergence rate of y to the ESS point (1,1) is accelerated as the value of C2 goes down from 0.5 to 0.1. The simulation results show that reducing the cost of social capital's strategy of 'innovation' is helpful to enhance the willingness of social capital to carry out low-carbon technology innovation in PPP projects. In Figs 5 and 6, we can see that the convergence rate of y to the ESS point (1,1) is accelerated as the value of L1 and S1 go bigger. The simulation results show that increasing the punishment of 'non-innovation' and the subsidies of 'innovation' are helpful to enhance the willingness of social capital to carry out low-carbon technology innovation in PPP projects.
The influence of C1 and L3 changes on the evolutionary of government's behavior. The evolutionary trend between government and social capital under the initial values of each parameter is shown as blue curves in Figs 7(A) and 8(A). Keep the other parameters unchanged, we can observe the evolutionary trend of government's behavior by changing the values of C1 and L3. In Fig 7(B), we can see that the convergence rate of x to the ESS point (1,1,1) is accelerated as the value of C1 goes down. The simulation results show that reducing the cost of government's strategy of 'regulation' is helpful to enhance the willingness of government to promote low-carbon technology innovation in PPP projects. In Fig 8(B), we can see that the convergence rate of x to the ESS point (1,1,1) is accelerated as the value of L3 goes bigger. The simulation results show that increasing the punishment from superior supervisory authority will help increase the willingness of government to promote low-carbon technology innovation in PPP projects.
The influence of C2, S1, L1 and L4 changes on the evolutionary of social capital's behavior. The evolutionary trend between government and social capital under the initial values of Keep the other parameters unchanged, we can observe the evolutionary trend social capital's behavior by changing the values of C2, S1, L1 and L4. In Fig 9(B), we can see that the convergence rate of y to the ESS point (1,1,1) is accelerated as the value of C2 goes down from 0.5 to 0.1. The simulation results show that reducing the cost of 'innovation' is helpful to enhance the willingness of social capital to carry out low-carbon technology innovation in PPP projects. In Fig 10(B), we can see that the convergence rate of y to the ESS point (1,1,1) is accelerated as the value of S1 goes bigger from 0.1 to 0.5. The simulation results show that increasing the subsidies of 'innovation' is helpful to enhance the willingness of social capitals to carry out low-carbon technology innovation in PPP projects. In Figs 11(B) and 12(B), we can see that the convergence rate of y to the ESS point (1,1,1) is accelerated as the value of L1 and L4 go bigger 1 to 5. The simulation results show that increasing the punishment of 'non-innovation' and the compensation from social capital to consumer are helpful to enhance the willingness of social capital to carry out low-carbon technology innovation in PPP projects.
The influence of C3 and S2 changes on the evolutionary of consumer's behavior.. The evolutionary trend between government and social capital under the initial values of each parameter is shown as blue curves in Figs 13(a) and 14(a). Keep the other parameters unchanged, we can observe the evolutionary trend of consumer's behavior by changing the values of C3 and S2. In Fig 13(B), we can see that the convergence rate of z to the ESS point (1,1,1) is accelerated as the value of C3 goes down from 0.5 to 0.1. The simulation results show that reducing the cost of consumer's strategy of 'consumption' is helpful to enhance the willingness of consumer to buy low-carbon PPP products. In Fig 14(B), we can see that the convergence rate of z to the ESS point (1,1,1) is accelerated as the value of S2 goes bigger from 0.1 to 0.5. The simulation results show that increasing the subsidies of 'consumption' will help increase the willingness of consumer to buy low-carbon PPP products.

Discussion
In this study, it is found that the ultimate payer under different payment mechanisms of PPP projects are not the same, which is different from previous studies based on consumer payment [21]. So the subjects and influencing factors of low-carbon technology innovation diffusion under the two payment mechanisms are also different. Previous research results can not well reflect the evolutionary mechanism of low-carbon technology innovation diffusion of PPP projects in China.
In the existing research results, it is concluded that government is an important promoter of low-carbon technology diffusion, and its guiding policies and attitudes in low-carbon technology innovation will play a significant role [14,17]. But under government payment mechanism, government that shoulders the social responsibility of low-carbon public goods and lowcarbon transformation of the construction industry is not only an important subject to promote low-carbon technology innovation of PPP projects, but also the payer of PPP products [22]. The regulation cost of the government's promotion of low-carbon technology innovation and the punishment of its non-regulation behavior by superior supervisory authority are the important factors affecting its behavior strategy. We can improve the supervision efficiency of superior supervisory authority on the government's promotion of low-carbon technology innovation, and further enrich regulation means with the help of the media and the public. The government's willingness to assume low-carbon technology innovation should be enhanced by increasing penalties for non-regulation behaviors. Although government is no longer the payer of PPP products under user payment mechanism, it still bears the main responsibility of promoting low-carbon technology innovation, and its regulation behavior plays an important role in promoting the evolutionary of social capital and consumer behavior [18].
No matter under which mechanism, social capital is the implementation subject of low-carbon technology innovation. Existing studies have demonstrated that it is difficult to promote the diffusion of low-carbon technology completely depending on their own will, and the regulatory policies of government and consumer's consumption will have a huge impact on their behavior [19]. Through simulation, it is found that the cost of low-carbon technology innovation, the intensity of subsidy and the punishment from the government are the main factors affecting the behavior strategy of social capital. We can increase the government's punishment and subsidy for non-innovation of social capital. Awareness of low-carbon technology innovation in social capital should be improved by means of publicity and training, and gradually changed from institutional promotion to habitual adoption.
Under user payment mechanism, consumers become the ultimate payers of PPP products and services, and their consumption willingness plays an important role in promoting PPP low-carbon technology innovation. Through simulation, it is found that consumers' income from purchasing low-carbon products and government subsidies are the key factors influencing their choice of behavioral strategies. We can enhance consumers' awareness of low-carbon economy and enhance their awareness of low-carbon consumption by increasing social publicity. By increasing the government's guidance and subsidies for consumers to purchase lowcarbon products, it can promote the formation of consumers' low-carbon consumption intention.

Conclusion
This study investigates the diffusion of low-carbon technology innovation in PPP projects under different payment mechanisms (government payment mechanism and consumer payment mechanism). It is found that under government payment mechanism, the regulation cost of the government's promotion of low-carbon technology innovation and the punishment of its non-regulation behavior by superior supervisory authority are the important factors affecting its behavior strategy, and the cost of low-carbon technology innovation, the intensity of subsidy and the punishment from the government are the main factors affecting the behavior strategy of social capital. Under consumer payment mechanism, consumers' income from  purchasing low-carbon products and government subsidies are the key factors influencing their choice of behavioral strategies.
In view of the unique payment mechanism of PPP projects, this study employs evolutionary game theory to build the evolutionary game models under two payment mechanisms. The behavior evolutionary mechanism of different stakeholders under the two mechanisms is discussed respectively. The evolutionary relationship between the choice of behavior strategies   and the change of influencing factors under two mechanisms is analyzed by numerical simulation. The research results enrich the research perspective of low-carbon technology innovation diffusion and the application of evolutionary game theory on PPP projects. Furthermore, this study has some practical implications for managers of PPP projects to promote the diffusion of low-carbon technology innovation in PPP projects and accelerate the sustainable development of infrastructure industry.   However, there remain some limitations. Evolutionary game theory used in this study is mainly based on the expected utility theory in which the individual perceived value are not considered. Individual behavior is not only completely following the utility maximization but also will be affected by a variety of psychological factors [58]. Therefore, it is not realistic to use expected utility to measure the evolutionary of each stakeholder's decision-making behavior and perceived value will be considered in our future research. In addition, the choice of stakeholder's behavior will not only be affected by its own conditions, but also by the network environment in which it is embedded [63], the authors would discuss the diffusion of low-carbon technology innovation of PPP projects from the perspective of network in the future.